The functions \(\hat{\rho }_n\)  are easily seen to be \(|G|\)  invariant and hence descend to functions \(\rho _n\colon |M|\rightarrow I\) .
Let \(U_n = \hat{\rho }_n^{-1}(0,1]\)  and let \(V_n = \rho _n^{-1}(0,1]\)  so that
\(U_n = |\pi |^{-1}V_n\)  (and hence \(U_n\)  is \(|G|\)  invariant).
Following [1]}, [2]} let \(r_n\colon |P|\rightarrow |P|\)  denote the map \(r_n(|x,t|) = h_n(|x,t|,1)\) .
Then we have (see [1]}, [2]}) the following chain of inclusions
\(|P|_n \setminus |P|_{n-1} \subset U_n \subset r_n^{-1}(|P|_n \setminus |P|_{n-1})\) 
